What a great question!
Adding new content on a new page to a redesigned site is pretty safe right off the get-go, especially now that Google has indexed your site.
If you are adding new content to an already indexed page, you should be ok to proceed at this point as well.
Where it gets a little trickier is when you are refreshing or outright changing established content. That is where you want to give the search engines time to catch up, reindex your domain, and do a check on what keywords in that content you are ranking well for. If a page is performing strong for certain keywords, you will want to watch for longer periods of time to read the trends before making drastic changes. If you are seeing a dip, then you can mindfully start shifting your content.
You won’t see much within the first month, especially if there are changes in the regular search cycle (like now with the COVID-19 disruption).